Properly immersed surfaces in hyperbolic 3-manifolds

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We study complete finite topology immersed surfaces Σ in complete Riemannian 3-manifolds N with sectional curvature Kₙ≤ -a²≤ 0, such that the absolute mean curvature function of Σ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface Σ must be proper in N and its total curvature must be equal to 2πχ(Σ). If N is a hyperbolic 3-manifold of finite volume and Σ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of Σ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in N.